# What is the Difference Between Inscribed and Intercepted Arc? Your Guide to Understanding Arcs in Geometry

When it comes to arcs, whether it be in geometry or the technical aspects of mechanics and dynamics, there are several types of arcs that many people come across. These include circular, semicircular, major, minor, and the two types that often get mistaken for the other. These two types are inscribed and intercepted arcs, which can be easily confused when dealing with circles and their corollary properties.

So what exactly is the difference between these two types of arcs? In brief, an intercepted arc refers to a segment of the circumference of a circle that is intercepted by an angle of the circle. An inscribed arc, on the other hand, is the length of an arc that lies along the circumference of a circle that is inscribed within a given angle. While it might sound like a subtle difference between the two, it can have a significant impact on mathematical calculations, which is why it is essential to understand them.

Another critical factor to keep in mind when exploring the world of arcs is to understand their relationship to angles within a circle. This is because an angle that is inscribed in a circle’s circumference will have an intercepted arc that is twice the size of the arc resulting from the angle’s diameter. This information can play an essential role in various mathematical formulas and calculations, making it all the more crucial to have a clear understanding of inscribed and intercepted arcs’ differences.

## Definition of inscribed arc vs intercepted arc

Understanding the concept of inscribed and intercepted arcs is crucial in geometry. These terms deal with arcs in circles and their relationship with one another. Let us dive deeper into the definitions of inscribed arc vs intercepted arc.

• An inscribed arc is an arc that lies entirely inside a circle, and its endpoints are on the circle. In simpler terms, it is an arc formed by connecting two points on a circle’s circumference. Another crucial feature of inscribed arcs is that they subtend an angle at the circle’s center. This angle is equal to the arc’s degrees.
• On the other hand, an intercepted arc is an arc that lies partially inside and partially outside the circle. It is formed when a line intersects a circle at two points, and the arc between these points is the intercepted arc. In simpler terms, it is the part of a circle’s circumference that lies in between two points on the same line.

## Key Properties of Inscribed Arc and Intercepted Arc

When dealing with circles, it is important to understand the difference between an inscribed arc and an intercepted arc. Both types of arcs play a key role in circle geometry and have unique properties.

## Properties of Inscribed Arc and Intercepted Arc

• Inscribed Arc: An inscribed arc is an arc that lies completely within a circle and passes through two points on the circle. The key property of an inscribed arc is that its central angle is twice the measure of the arc itself. This means that if you know the measure of the arc, you can easily find the measure of the central angle by simply doubling it.
• Intercepted Arc: An intercepted arc is an arc that is formed when two chords intersect within a circle. The key property of an intercepted arc is that it has an angle measure equal to half the sum of the measures of the two angles that intercept it.

## Properties of Inscribed Arc

When dealing with inscribed arcs, there are a few key properties to keep in mind:

• The measure of an inscribed angle is half the measure of its intercepted arc.
• If two inscribed angles intercept the same arc, then their measures are equal.
• The sum of the measures of two inscribed angles that intersect inside a circle is equal to the measure of the intercepted arc they create.

## Properties of Intercepted Arc

When dealing with intercepted arcs, there are a few key properties to keep in mind:

• If two chords intersect inside a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord.
• If a tangent and a chord intersect at a point on a circle, then the measure of the angle formed is equal to half the measure of the intercepted arc.
Property Diagram
If two chords intersect inside a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord.
If a tangent and a chord intersect at a point on a circle, then the measure of the angle formed is equal to half the measure of the intercepted arc.

## Geometrical relationships involving inscribed and intercepted arc

Understanding the geometrical relationships involving inscribed and intercepted arc is essential in geometry. Here are three subtopics to help you understand the difference between inscribed and intercepted arc.

## Inscribed Arc

• An inscribed arc is an arc that lies on the interior of a circle.
• The arc is inscribed in a circle if its endpoints lie on the circumference of the circle.
• An angle formed by two chords that intersect inside a circle is half the sum of the intercepted arcs.

## Intercepted Arc

• An intercepted arc is an arc that lies inside a circle and is intercepted by an angle.
• The intercepted arc is that portion of the circle that lies between the endpoints of the angle.
• When two secants are drawn to a circle, the product of the intercepted segments on one is equal to the product of the intercepted segments on the other.

## The Relationships

There are several relationships that exist between inscribed and intercepted arc in a circle:

• Two arcs that are inscribed in the same circle are congruent if and only if their corresponding central angles are congruent.
• Two angles that intercept the same arc are congruent.
• The measure of an inscribed angle is half the measure of its intercepted arc.

## Examples of Geometrical Relationships Involving Inscribed and Intercepted Arc

Here is an example of how geometric relationships involving inscribed and intercepted arc can be used to solve a problem:

 Given that AC and BD are diameters of the same circle, find the measure of angle ACD. Since AC and BD are diameters, they intersect at the center of the circle, O. Therefore, angle ACD is half of the intercepted arc AD. Since AC is a diameter, angle ADC is a right angle. Therefore, intercepted arc AD is one-half of the circle. So, m(arc AD) = 1/2(360°) = 180°. Therefore, m(angle ACD) = 1/2m(arc AD) = 1/2(180°) = 90°.

## Applications of inscribed and intercepted arc in real-life scenarios

Both inscribed and intercepted arcs are important mathematical concepts that have several real-life applications. Here are some of the ways they are used:

• Architecture and Engineering: Architects and engineers use inscribed and intercepted arcs to calculate dimensions and angles while designing buildings and structures. For example, the diameter of a circular window can be determined by measuring the inscribed angle from the window’s edges. Similarly, the position of columns and pillars can be determined by calculating the intercepted angle between them.
• Land Surveying: In land surveying, inscribed and intercepted arcs are used to measure and calculate angles and distances. Surveyors use a theodolite, a specialized instrument that can measure vertical and horizontal angles, to determine the location of points on a map. The measurements of angles and distances are then used to calculate the area and dimensions of the land under survey.
• Astronomy: Inscribed and intercepted arcs are used in astronomy to determine the position and movement of celestial objects. The apparent motion of the stars and planets can be tracked by measuring the intercepted arc of their movement across the sky. This information is used to create star maps and to navigate ships and planes.
• Mechanical Engineering: The principle of inscribed and intercepted arcs is used in mechanical engineering to design gears, camshafts, and other mechanical systems. The teeth of a gear are designed to fit in an inscribed circle, and the angles of the teeth are calculated using intercepted arcs. By controlling the size and angles of the teeth, mechanical engineers can optimize the power, speed, and efficiency of the system.

The applications of inscribed and intercepted arcs are numerous and diverse. From architecture to astronomy, these mathematical concepts play a vital role in many aspects of our modern world.

Table:

Real-life Scenario Use of Inscribed Arc Use of Intercepted Arc
Architecture and Engineering Determine dimensions of circular objects Determine position and angle of objects
Land Surveying Calculate area and dimensions of land Measure and calculate angles and distances
Astronomy Create star maps and navigate ships and planes Determine the position and movement of celestial objects
Mechanical Engineering Design gears to optimize power and efficiency Control speed and movement of mechanical systems

The uses of inscribed and intercepted arcs are varied and far-reaching, offering valuable insights to a wide range of industries and disciplines. By understanding these principles, we can solve mathematical problems, design innovative solutions, and navigate our complex, interconnected world with greater precision and clarity.

## How to find the measure of inscribed and intercepted arc

Knowing how to find the measure of inscribed and intercepted arc is an essential skill for students studying geometry. Here are the steps to finding the measure of the arcs:

• Finding the measure of an inscribed arc: An inscribed arc is the arc that lies within the interior of a circle and whose endpoints are on the circle. To find the measure of an inscribed arc, one must know the central angle that subtends the arc. The central angle is twice the size of the inscribed angle. Use the formula: measure of inscribed angle = 0.5 x measure of central angle.
• Finding the measure of an intercepted arc: An intercepted arc is the arc that lies between two intersecting chords of a circle that share the same endpoint. To find the measure of an intercepted arc, one must know the measure of the two corresponding inscribed angles that share the same intercepting arc. Then, use the formula: measure of intercepted arc = 2 x measure of one of the corresponding inscribed angles.

Let’s take a closer look at the process of finding the measure of inscribed and intercepted arcs:

First, let’s assume we have a circle with center O:

Next, let’s assume we have a chord AB and an inscribed angle AOB that subtends an arc ACB:

The measure of the central angle AOC that subtends the same arc ACB can be found using the formula:

measure of central angle = 2 x measure of inscribed angle

Therefore:

measure of central angle AOC = 2 x measure of inscribed angle AOB

Now that we know the measure of central angle AOC, we can use the formula to find the measure of the intercepted arc ACB:

measure of intercepted arc ACB = 2 x measure of inscribed angle AOB

Using the same circle with center O and the chords AB and CD that intersect at point E, we can find the measure of the intercepted arc FG that is formed by the two corresponding inscribed angles:

Object Formula Measurement
Measure of inscribed angle EAF 0.5 x measure of arc EF 75°
Measure of inscribed angle DCF 0.5 x measure of arc FG 110°
Measure of intercepted arc FG 2 x measure of inscribed angle EAF 150°

Therefore, the measure of intercepted arc FG is 150°.

Knowing how to find the measure of inscribed and intercepted arcs is essential in solving problems involving circles and their properties. With practice, students will be able to apply these concepts in their geometry studies and beyond.

## Formula for calculating inscribed and intercepted arc

When dealing with circles, it is important to understand the difference between inscribed and intercepted arcs. In a circle, an inscribed arc is an arc that lies inside the circle and touches both ends of a chord. An intercepted arc, on the other hand, is an arc that lies outside the circle and is bounded by two intersecting chords or a chord and a tangent line.

Calculating the measure of inscribed and intercepted arcs can be done using the following formulas:

• The measure of an inscribed arc is half the measure of the central angle that subtends the arc. This can be represented by the formula: measure of inscribed arc = 1/2 x measure of central angle
• The measure of an intercepted arc is the difference between the measures of the two inscribed angles that intersect the arc. This can be represented by the formula: measure of intercepted arc = measure of arc of larger inscribed angle – measure of arc of smaller inscribed angle

To better understand these formulas, let’s take a look at an example:

Measure of central angle (θ) Measure of inscribed arc Measure of intercepted arc
60 degrees 30 degrees 60 degrees (since the two inscribed angles are equal)
120 degrees 60 degrees 120 degrees (since the two inscribed angles are equal)
150 degrees 75 degrees 105 degrees (since the arc is bounded by two inscribed angles of 75 and 30 degrees)

By using these formulas, you can easily calculate the measures of inscribed and intercepted arcs in a circle. This knowledge can be helpful in a variety of situations, such as when working with angles in geometry or determining the distance between two points along a curved path.

## Theorems Related to Inscribed and Intercepted Arc in Geometry

When we talk about circles in geometry, there are two important concepts that often come up – inscribed arc and intercepted arc. To understand these concepts better, we must first define what they mean:

• Inscribed arc: Also known as a minor arc, this is an arc of a circle that lies within an angle formed by two intersecting chords of the circle.
• Intercepted arc: Also known as a major arc, this is an arc of a circle that lies between two intersecting chords of the circle and contains the vertex of the angle formed by the two chords.

Now that we know what inscribed and intercepted arcs are, we can look at some of the theorems related to them:

Theorem 1: An angle inscribed in a circle is half of the central angle that intercepts the same arc.

This theorem tells us that if we have an angle inscribed in a circle and another angle that intercepts the same arc, the inscribed angle will always be half the size of the central angle. Take a look at the diagram below for a better understanding:

Theorem 2: If two inscribed angles of a circle or congruent, then the intercepted arcs are congruent.

Put simply, if we have two inscribed angles of a circle that are the same size, the arcs they intercept will also be the same size. This is shown in the diagram below:

Theorem 3: If a diameter of a circle bisects a chord, then it is perpendicular to the chord.

In this theorem, we are looking at the relationship between the diameter of a circle and a chord. If a diameter bisects a chord (i.e. it divides the chord into two equal parts), then it will always be perpendicular to the chord. This can be seen in the diagram below:

Theorem 4: Description:
If two chords in a circle are congruent, then their intercepted arcs are congruent. This theorem tells us that if two chords in a circle are the same length, then the arcs they intercept will also be the same length.
If a tangent and a chord intersect at a point on a circle, then the intercepted arc is bisected by the chord. This theorem tells us that if a tangent and a chord intersect at a point on the circle, then the chord will always bisect the intercepted arc.
Two tangents drawn to a circle from the same point are congruent. If we draw two tangents from the same point outside a circle, we can prove that they are equal in length. This theorem is useful in solving many circle-related problems.

Theorem 5: If a radius of a circle is perpendicular to a chord, then it bisects the chord.

Similar to Theorem 3, this theorem tells us that if a radius of a circle is perpendicular to a chord, then it will always bisect the chord into two equal parts. The diagram below illustrates this concept:

Understanding the relationships between inscribed and intercepted arcs in geometry is crucial in solving a variety of circle-related problems. These theorems are just a few examples of the powerful and elegant concepts involved.

## What is the difference between inscribed and intercepted arc?

Q: What is an inscribed arc?
An inscribed arc is an arc that lies inside a circle and starts and ends on the circle’s circumference.

Q: What is an intercepted arc?
An intercepted arc is an arc that lies inside the circle and is formed by two intersecting chords.

Q: How are inscribed and intercepted arcs different?
The main difference between an inscribed and intercepted arc is that an inscribed arc starts and ends on the circle’s circumference, while an intercepted arc is formed by two chords that intersect inside the circle.

Q: What are some properties of inscribed arcs?
Inscribed arcs are equal to half of the circle’s circumference if they span an entire diameter. They are also congruent to each other if their corresponding chords are congruent.

Q: What are some properties of intercepted arcs?
Intercepted arcs that have the same intercepted angle are congruent. Also, the sum of two intercepted arcs that form a larger arc is equal to the length of the larger arc.